1. Prime Numbers The Fundamental Theorem of Arithmetic
Every whole number can be factored uniquely into a product of prime number powers.
For example, =3*3*3607*3803=32*3607*3803
Prime numbers are the code for writing all whole numbers.
But factoring or decoding into primes is hard.

2. Prime Numbers and Pretty Good Privacy

3. Pretty good privacy
PGP is a tongue in cheek expression for an encryption scheme considered nearly impossible to break.
Use public key encryption based on products of large prime numbers

4. Euclidean algorithm Eudoxus of Cnidus (about 375 BC),
function gcd(a, b)
if a = 0 return b
while b ≠ 0
if a > b a := a − b
else b := b − a
return a

5. Example: Find GCD(24,15) a is not zero; a=24, b=15
Reset a-> a-b=25-15=9
Now 9<15 so reset b-> 15-9=6
Now a=9>b=6 so reset a-> 9-6=3
Now b=6>a=3 so reset b->6-3=3
Now a=b so return a=3.

6. Euclid’s algorithm: GCD
The greatest common divisor of M and N is the largest whole number that divides evenly into both M and N
GCD (6 , 15 ) = 3
If GCD (M, N) = 1 then M and N are called relatively prime.
Euclid’s algorithm: method to find GCD (M,N)

7. Euclid’s algorithm M and N whole numbers.
Suppose M ≤ N. If N is divisible by M then GCD(M,N) = M.
Otherwise, subtract from N the biggest multiple of M that is smaller than N. Call the remainder R.
N=MK+R or R=MK-N. If Q divides into both M and N then Q divides into R. So:
GCD(M,N) = GCD (M,R).
Repeat until R divides into previous.

8. Example: GCD (105, 77) 77 does not divide 105.
Subtract 1*77 from 105. Get R=28
28 does not divide into 77. Subtract 2*28 from 77. Get R=77-56=21
Subtract 21 from 28. Get 7.
7 divides into 21. Done.
GCD (105, 77) = 7.

9. Example: GCD (105, 47) 47 does not divide 105.
Subtract 2*47 from 105. Get R=11
11 does not divide into 47. Subtract 4*11 from 47. Get R=47-44=3
3 does not divide 11. Subtract 3*3 from 11. R=2
2 does not divide 3. Subtract 2 from 3. R=1
GCD (105, 47) = 1.

10. Another example gcd(1071,1029) =gcd(1029,42) (42= 1071 mod 1029
=21: since b=0, we return a

11. Run time of Euclidean (O(N^2)). Red: fast, blue: slow

12. Clicker Exercise: find GCD (1221,121)
The GCG of 1221 and 121 is:
A) 2
B) 21
C) 11
D) 121
E) 1

13. [GCD(6251,1473)=]
[1]
[3]
[7]
[11]

14. Relatively prime
Two numbers M and N are called relatively prime if GCD(M,N)=1.
Example: Any prime number is relatively prime to any number other than itself. GCD(11,9)=1
Example: Powers of different primes are relatively prime to one another. GCD(9,16)=1
Two numbers are relatively prime iff their prime factorizations are distinct.

15. Prime numbers
A whole number is called prime if it is relatively prime to every smaller whole number.

16. Prime factorization theorem
every natural number > 1 can be written as a unique product of prime numbers.
E.G: 6936=2 x 2 x 2 x 3 x 17 x 17=2^3 x 3 x 17^2
6936 ≠ any other product of prime powers
practically proved by Euclid,
first “ correct” proof in Disquisitiones Arithmeticae by Gauss.

17. Large prime numbers Euclid: infinitely many prime numbers
Proof: given a list of prime numbers, multiply all of them together and add one.
This new number is not divisible by any number on our list.
So either the new number is prime, or it is divisible by a prime not on the list.

18. Euclid’s proof
Consider any finite set of primes. Multiply all of them together and add 1 (see Euclid number). Call this Q
Dividing Q by any of these would give a remainder of 1.
So Q is not divisible by any number in this list.
Any non-prime can be decomposed into a product of primes,
Either Q is prime itself, or there is a prime number in the decomposition of Q that is not in the original finite set of primes.
Either way, there is at least one more prime that was not in the finite set we started with. This argument applies no matter what finite set we began with. So there are more primes than any given finite number.

19. Infinitude of primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 …
list of prime numbers

20. Testing for prime numbers
Is 97 a prime number?
How about 111?
How about ?

21. Finding all prime numbers up to a given size
Sieve of Eratosthenes: Make a square whose side is at least the square root of the given number. Cross out all multiples of two, then all multiples of three, etc.

22. Group work
Use the sieve of Eratosthenes to compute all primes from 1 to 480. Note that 22*22=484 so you only need to consider all multiples of primes up to 19 which is the largest prime less than 22.

23. Computers can factor small prime numbers

24. Primality test algorithm to determine if “N is prime.”
Factorization is hard; primality testing is easy.
elliptic curve primality test O((log n)^6),
Log(n) is, approximately, the number of digits that n has.
The largest known prime has 17 million digits. Raising this to the 6th power gives a number with about 40 digits. If a computer can execute a trillion operations per second, we re talking on the order of 10^22 seconds. We are talking on the order of quadrillions of years.

25. RSA 200
RSA-200 =
RSA-200 = ×

26. How long to factor RSA 200?
If a k-digit number is the product of two primes
No known algorithm can factor in polynomial time, i.e., that can factor it in time O(k^p) for some constant p.
There are algorithms faster than O((1+ε)^k) i.e., sub-exponential.
For a quantum computer, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time O(N^3) and O(N) memory.
In 2001, the first 7-qubit quantum computer became the first to run Shor's algorithm. It factored the number 15
GNFS: O(2^(N^(1/3))

27. RSA competitions
German Federal Agency for Information Technology Security (BSI) team
On May 9, 2005, factored RSA-200, a 663-bit number (200 decimal digits), using the general number field sieve.
…later: RSA-640, a smaller number containing 193 decimal digits (640 bits), on November 4, 2005.
Both factorizations required several months of computer time using the combined power of 80 AMD Opteron CPUs.

28. [The number 2*3*5*7*11+1=2311 is prime]
True
False

29. There are lots of primes known to man
Prime number theorem: the number of primes less than or equal to N is on the order of N divided by log N.

30. Largest known prime
257,885,161 − 1 (2013)
243,112,609 − 1. (2008)

31. RSA

32. RSA history
Algorithm described in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT;
Clifford Cocks, a British mathematician working for the UK intelligence agency GCHQ, described an equivalent system in an internal document in His discovery, however, was not revealed until 1997 due to its top-secret classification, and Rivest, Shamir, and Adleman devised RSA independently of Cocks' work.
MIT was granted U.S. Patent 4,405,829 for a "Cryptographic communications system and method" that used the algorithm in The patent would have expired in 2003, but was released to the public domain by RSA on 21 September 2000.

34. Mathematical Cryptography
W.S. Jevons (1835—1882)
The Principles of Science:
A Treatise on Logic and Scientific Method (1890s)
'direct' is “easy,” but ‘inverse’ is ‘hard’.
encryption is easy; decryption is hard.
Multiplication: easy, factoring: hard.
Jevons anticipated RSA Algorithm

35. Simple; multiply numbers
Difficult: factor numbers.
example x 99991= (easy)
M= = A xB
Find A and B (difficult)
Computer: check primes up to square-root (roughly 38000).

36. The RSA encryption algorithm
N=PQ (product of two primes)
Φ(N) = (P-1)(Q-1)
Encryption key: 1<E<φ(N) such that
GCD(E , φ) = 1
Decryption key: D such that
DE ≡ 1 mod (φ)
M< φ

37. Encryption/Decryption
C=ME mod (N)
R=CD mod (N)
CLAIM: R=M (the original message)

38. Short digression: modular arithmetic
A ≡ B mod (C)
Means that B is the remainder when C is divided into A
For example, 13 ≡ 1 mod (12)
If it is 3:30 now then in 13 hours it will be 4:30.
Shorthand: B=mod(A,C)
Arithmetic:
mod (MN, C)=mod(mod(M,C) x mod(N,C), C))

39. Laws of exponents Regular exponents Modular exponents (ab)c = ac bc
mod((ab)c ,m)=
mod(mod (ac ,m) mod (bc ,m), m)
ab+c=ab ac
mod(ab+c,m)=
mod(mod (ab ,m) mod (ac ,m), m)
(ab)c = abc
mod((mod (ab ,m))c, m)

40. Examples mod((13)25 ,12)= mod((mod (13 ,12))25, 12)=
mod(225, 12)= mod(mod(24, 12)6 x mod(2,12), 12)=
mod(mod(4, 12)5 x2, 12)=mod(8 , 12)=8

41. Proof of RSA C=ME mod (N) R=CDmod (N) = (MD mod (N))E mod (N) =
(MDE mod (N))
DE ≡ 1 mod (N)
(M1 mod (N)) =M (since M< N)
Fermat’s little theorem: aP-1 ≡ 1 mod (P)

42. Plaintext to numbers - A B C D E F G H I J K L M N O P Q R S T U V W XY Z 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

43. Plaintext message numerical version Kill Bill
Note: M<φ so may need to send message in pieces, e.g. one letter at a time

44. Example
N = 5 x 7 =35
Φ=4x6=24
E=11 then GCD (E, Φ)=1
D=11 then DxE=121=5x24+1
So DxE≡1 mod 24
In this case decryption is just the inverse of encryption because E=D. Generally note true.

45. - A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 28 27 33 32 30 31

46. EXERCISE
Use the cipher table above to decrypt the following ciphertext into plain text:

47. Solution:
Flattery will get you nowhere, but don't stop trying

48. How long to factor large products?
what if the number to be factored is not ten digits, but rather 400 digits?
square-root : 200 digits.
lifetime of universe: approx. 10^{18} seconds.
If computer could test one trillion factorizations per second, in the lifetime of the universe it could check 10^{30} possibilities.
But there are 10^{200} possibilities.

49. Pretty Good Privacy

50. PGP Based on “public key” cryptography
Binds public key to user name or address
Authentication: “digital signature” used to verify identity of sender
integrity checking: used to detect whether a message has been altered since it was completed
Encryption: based on RSA/DSA
Decryption based on public key
Web of trust: third party vetting

51. Zimmerman, 1992 As time goes on…
you accumulate keys from other “trusted” parties.
Others each choose their own trusted parties.
everyone gradually accumulates and distributes with their key certifying signatures from others
Expectation: anyone receiving it will trust at least one or two of the signatures.
emergence of a decentralized fault-tolerant web of confidence for all public keys.

52. Any agency wanting to read PGP messages would probably use easier means than standard cryptanalysis,
e.g. rubber-hose cryptanalysis or black-bag cryptanalysis i.e. installing some form of trojan horse or keystroke logging software/hardware on the target computer to capture encrypted keyrings and their passwords.
The FBI has used this attack against PGP.
such vulnerabilities apply to any encryption software.

53. Criminal investigation of Zimmerman
PGP encryption found its way outside the US.
Cryptosystems using keys > 40 bits were considered munitions by US export regulations;
PGP keys >= 128 bits.
Feb 1993: Zimmermann targeted by US Govt for "munitions export without a license".
Penalties … were substantial.
Zimmermann challenged these regulations in a curious way.
Published PGP source code as hardback book (MIT Press)
To build: buy the $60 book, scan pages using an OCR program, GNU C Compiler. PGP would thus be available anywhere in the world.
Export of munitions restricted; export of books is protected ( First Amendment).
US export regulations regarding cryptography remain in force, but were liberalized substantially …. PGP … can be exported internationally except to 7 specific countries and a named list of groups and individuals.

54. The future of cryptography?
As of 2005, the largest number factored by a general-purpose factoring algorithm was 663 bits long (see RSA-200), using a state-of-the-art distributed implementation. RSA keys are typically 1024–2048 bits long. Some experts believe that 1024-bit keys may become breakable in the near term (though this is disputed); few see any way that 4096-bit keys could be broken in the foreseeable future. Therefore, it is generally presumed that RSA is secure if n is sufficiently large. If n is 256 bits or shorter, it can be factored in a few hours on a personal computer, using software already freely available. Keys of 512 bits (or less) have been shown to be practically breakable in 1999 when RSA-155 was factored by using several hundred computers. A theoretical hardware device named TWIRL and described by Shamir and Tromer in 2003 called into question the security of 1024 bit keys. It is currently recommended that n be at least 2048 bits long.

55. Is RSA safe?
In 1994, Peter Shor published Shor's algorithm, showing that a quantum computer could in principle perform the factorization in polynomial time. However, quantum computation is still in the early stages of development and may never prove to be practical.